Good to Know

Some new stuff on the origins of humanity and primates and the Earth’s long-term climate, and rock technology, and a recommendation for a British Museum podcast on 100 items from their collection, and an extinct pig the size of a small hippo. Check it.

Thorough Knowledge

I don’t know what I’m looking for. I suppose that I want to have a better grasp on how humans went from bands of apes to talking on the telephone.

Twas the Cenozoic Era…, (meaning the current geologic era which started 66 million years ago- when the dinosaurs died, so I should say, “Tis the Cenozoic Era” really). During the Eocene epoch, (55-35 mya), primates, similar to lemurs, emerge in the northern hemisphere during a period of global warming to secure daytime openings in the food-chain. * Then in the following epoch, the Oligocene (~35-22 mya), the presence of these prosimian primates fades in the less-forested areas throughout the world as the overall climate cooled, but they continue to exist in Africa.  During the next epoch, the Miocene (23-5 mya), the climate cooled further and forests declined. At this time, monkeys spread and branched out. About 20 mya…

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As for actual numbers,

narmermacehead“The Narmer macehead is an ancient Egyptian decorative stone mace head. It was found during a dig at Kom al Akhmar, the site of Hierakonpolis (ancient Egyptian Nekhen.) It is dated to the reign of king Narmer whose serekh is engraved on it. Today it is kept at the Ashmolean Museum, Oxford. Narmer was an ancient Egyptian pharaoh of the Early Dynastic Period (c. 32nd century BC). He is thought to be the successor to the Protodynastic pharaohs Scorpion (or Selk) and/or Ka, and he is considered by some to be the unifier of Egypt and founder of the First Dynasty, and therefore the first pharaoh of unified Egypt.”




The numerals occupy the center of the lower register. Four tadpoles below the ox, each meaning 100,000, record 400,000 oxen. The sky- lifting Heh- god behind the goat was the hieroglyph for “one million”; together with the four tadpoles and the two “10,000” fingers below the goat, and the double “1,000” lotus- stalk below the god, this makes 1,422,000 goats. To the right of these animal quantities, one tadpole and two fingers below the captive with his arms tied behind his back count 120,000 prisoners. These quantities makes Narmer’s mace the earliest surviving document with numbers from Egypt, and the earliest surviving document with such large numbers from anywhere on the planet.


“The quantities on Narmer’s Heb-Sed mace happen to combine, with good accuracy and in three closely related ways, two major mathematical constants that have intrigued many number researchers, phi and pi.

Some people claim these same constants were also embedded in the proportions of the Great Pyramid and other ancient Egyptian monuments, but many mainstream scholars assert those ratios got there by accident, without the builders’ knowledge or intent.”

From Wikipedia and “Ancient Creation Stories told by the Numbers”, by H. Peter Aleff on Recovered

For an explanation of Egyptian numbers, see “Rhind Papyrus“.


Counting seems so obvious. As infants, adults count our fingers and toes, the pieces of food on our trays, the colored blocks we play with. We learn to count from the beginning of our lives.

And then we move on to numbers. They must be memorized. We have symbols for 0 to 9 whose names and symbols must be memorized for immediate recall. It is so intrinsic a skill that we no longer consider their purpose or origin.

What did people do before symbolic language?

They made tick marks on sticks. That’s what they did. And that is such an effective method of counting that we still do it. We make groups of five marks. Four marks up and down and then a fifth across to complete the bunch.

The oldest known example of these kinds of marks is …

lebombo-boneThe Lebombo Bone.


The Lebombo bone is a baboon’s fibula with 29 distinct notches, discovered within the Border Cave in the Lebombo Mountains of Swaziland.


The number of notches suggests that the bone was used to mark the days of a lunar or menstrual calendar.

It has been dated to about 35,000 years ago (Middle Paleolith) in the excavation report of 1973. (Wikipedia; Pegg, Ed Jr. “Lebombo Bone.” From MathWorld; The oldest mathematical artefact by Bogoshi, Naidoo, and Webb)

Consider that it is thought that humans only began to migrate from Africa 50,000 years ago. 

The Lebombo bone is about as old as the oldest known piece of vigurative art, the Venus of Hohle Fels, found in southern Germany.

The Ishango Bone


The Ishango bone is similarly a fibula of a baboon but dating from much later, the Upper Paleolithic era, about 18,000 to 20,000 BC.ishmap

It has a series of tally marks carved in three columns running the length of the tool.

This gets a little detailed in the description of the markings, but it’s interesting to read how archaeologists attempt to decipher them.

There are three rows around the bone containing sets of tally marks:

First row: 19, 17, 13, 11

Second row: 7, 5, 5, 10, 8, 4, 6, 3

Third row: 9, 19, 21, 11




Some believe the three columns of asymmetrically grouped notches imply that the implement was used to construct a numeral system.


The central column begins with three notches, and then doubles to 6 notches.


The process is repeated for the number 4, which doubles to 8 notches, and then reversed for the number 10, which is halved to 5 notches.

These numbers may not be purely random and instead suggest some understanding of the principle of multiplication and division by two.

The bone may therefore have been used as a counting tool for simple mathematical procedures.

Furthermore, the numbers on both the left and right column are all odd numbers (9, 11, 13, 17, 19 and 21). The numbers in the left column are all of the prime numbers between 10 and 20 (which form a prime quadruplet), while those in the right column consist of 10 + 1, 10 − 1, 20 + 1 and 20 − 1. The numbers on each side column add up to 60, with the numbers in the central column adding up to 48. In the book How Mathematics Happened:The First 50,000 Years, Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC, with prime numbers probably not being understood until about 500 BC. He also writes that “no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10.”

(Wikipedia; Williams, Scott W.: “Mathematicians of the African Diaspora” The Mathematics Department of The State University of New York at Buffalo.; Rudman, Peter Strom (2007). How Mathematics Happened: The First 50,000 Years. Prometheus Books. p. 64.)

It’s just amazing to me to imagine a person sitting in the forest or next to a modest living space, and deciding that because he or she uses these numbers so often, or these numbers are so important, he needs to mark them on something sturdy so he can use it again. It isn’t incredibly easy to carve markings into a bone. It takes some intention. And a sharp tool.


From the Dictionary app on my mac:

A torture rack in the Tower of London – from Wikipedia

“usage: The relationship between the forms rack and wrack is complicated. The most common noun sense of rack, ‘a framework for holding and storing things,’ is always spelled rack, never wrack. In the phrase rack something up, the word is also always spelled rack. Figurative senses of the verb, deriving from the type of torture in which someone is stretched on a rack, can, however, be spelled either rack or wrack: thus, racked with guilt or wracked with guilt; rack your brains or wrack your brains. In addition, the phrase rack and ruin can also be spelled wrack and ruin .”

Rhind_Mathematical_PapyrusThe Rhind Papyrus is the best example of Egyptian mathematics and dates to  around 1650 BCE, but its contents date even further back. In its own introduction, the scribe, A’h-mose writes that he is copying material from 1849-1800 BCE. So four thousand years ago, the Egyptians were coming up with all sorts of fascinating materials. An excellent description of this papyrus is available in Eli Maor’s book, Trigonometric Delights, which is actually available online!

Egyptians wrote from right to left and used symbols to indicate numbers as shown below.


egyptian numbers copy


Maor, Eli (1998). Trigonometric DelightsPrinceton University Press. p. 20. ISBN 0-691-09541-8


This is Pascal’s Triangle. Do you see the pattern determining the next row? Basically, you add the two numbers above to get the next entry. So the next line would read: 1, 7, 21, 35, 35, 21, 7, 1.

For those who like math notation:

The formula for nCr is:

(The ! means factorial, aka: 5! = 1 x 2 x 3 x 4 x 5 = 120. It’s super useful in basic statistics, but that’s another entry.)

The weird thing about this is not how you make the pattern, but that the pattern is actually useful!

The sum of the rows gives the powers of 2.
Row 0: = 1 = 20
Row 1: = 1 + 1 = 2 = 21
Row 2: =  1 + 2 + 1 = 4 = 22
Row 3: = 1 + 3 + 3 + 1 = 8 = 23 …

If the second number in the row is prime, (because the first number in all the rows is 1, so skip it) the all of the numbers in that row are divisible by it. Check the 8th row that I wrote out up top as a good example:

1, 7, 21, 35, 35, 21, 7, 1

Something called the Magic 11’s:
If each row is a single number, it is a power of 11:
1 = 110
11 = 111
121 = 112
1331 = 113
14641 = 114  …

Here’s my favorite one. So let’s say you are trying to solve (x + y)n for some variables x and y, raised to the power n.
(x + y)0 = 1
(x + y)1 = x + y       aka 1x + 1y
(x + y)2 = x+ 2xy + y2
(x + y)3 = x3 + 3x2y + 3xy+ y3
(x + y)= x4 + 4x3y + 6x2y+ 4xy+ y4

The coefficients are the entries in the triangle! Crazy.

There’s a whole bunch more that’s probably only interesting to people who do math alot, but it’s pretty damn nifty. Check out for more.


In addition to being a word in English which contains all 5 of the vowels, it is also a word I didn’t know until I started reading The Iliad today. It’s contained in the foreword to my edition and it means, according to’s app…

1. Holding tenaciously to a purpose, course of action, or opinion; resolute
2. Stubborn or obstinate

From on pertinacious:

“Philosopher, mathematician, and writer, Bertrand Russell once said, “The whole problem with the world is that fools and fanatics are always so certain of themselves, and wiser people so full of doubts.”

This week’s words describe people who fall somewhere in that spectrum. Do any of them remind you of someone you know?…

“A man is pertinacious when he defends his folly and trusts too greatly in his own wit.” Geoffrey Chaucer; Canterbury Tales: Explicit Secunda Pars Penitentie; 1387-1400 (Translation: Walter W. Skeat). …


Seven blunders of the world that lead to violence: wealth without work, pleasure without conscience, knowledge without character, commerce without morality, science without humanity, worship without sacrifice, politics without principle. -Mahatma Gandhi (1869-1948)    ”

Thanks to Anu Garg for a great website!